A Hilbert-Nagata theorem in

noncommutative invariant theory

Authors:
Mátyás Domokos and Vesselin Drensky

Journal:
Trans. Amer. Math. Soc. **350** (1998), 2797-2811

MSC (1991):
Primary 16W20; Secondary 16R10, 16P40, 16W50, 13A50, 15A72

DOI:
https://doi.org/10.1090/S0002-9947-98-02208-9

MathSciNet review:
1475681

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Abstract: Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.

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Additional Information

**Mátyás Domokos**

Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O. Box 127, H-1364, Hungary

Email:
domokos@math-inst.hu

**Vesselin Drensky**

Affiliation:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria

Email:
drensky@math.acad.bg

DOI:
https://doi.org/10.1090/S0002-9947-98-02208-9

Keywords:
T-ideals,
algebras with polynomial identity,
noncommutative invariant theory,
algebra of invariants,
rational Hilbert series

Received by editor(s):
June 5, 1996

Additional Notes:
The first author was partially supported by Hungarian National Foundation for Scientific Research Grant no. F023436.

The second author was partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.

Article copyright:
© Copyright 1998
American Mathematical Society